Bacteria are often grown on agar plates and form circular colonies. the area of a colony is proportional to the number of bacteria it contains. the agar (a gelatinous substance obtained from red seaweed) is the resource that bacteria use to reproduce and so only those bacteria on the edge of the colony can produce new offspring. therefore the population dynamics are nt+1 = nt + i, where i is the input of new individuals and is proportional to the circumference of the colony, with proportionality constant r. let the area of the colony be directly proportional to the number of bacteria by a constant of proportionality k. also let k = 2π k π . (a) derive the recursion for the population dynamics. (write your answers in terms of n, nt, k, and r.)